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HARMONISING HARMONY

Published online by Cambridge University Press:  15 May 2015

LUCA TRANCHINI
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Abstract

The term ‘harmony’ refers to a condition that the rules governing a logical constant ought to satisfy in order to endow it with a proper meaning. Different characterizations of harmony have been proposed in the literature, some based on the inversion principle, others on normalization, others on conservativity. In this paper we discuss the prospects for showing how conservativity and normalization can be combined so to yield a criterion of harmony equivalent to the one based on the inversion principle: We conjecture that the rules for connectives obeying the inversion principle are conservative over normal deducibility. The plausibility of the conjecture depends in an essential way on how normality is characterized. In particular, a normal deduction should be understood as one which is irreducible, rather than as one which does not contain any maximal formula.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 8 , Issue 3 , September 2015 , pp. 411 - 423
Copyright
Copyright © Association for Symbolic Logic 2015 

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References

BIBLIOGRAPHY

Belnap, N. D. (1962). Tonk, plonk and plink. Analysis, 22(6), 130–134.CrossRefGoogle Scholar
Došen, K. (2003). Identity of proofs based on normalization and generality. Bulletin of Symbolic Logic, 9, 477–503.CrossRefGoogle Scholar
Dummett, M. (1981). Frege. Philosophy of Language (second edition). London: Duckworth.Google Scholar
Dummett, M. (1991). The Logical Basis of Metaphysics. London: Duckworth.Google Scholar
Ferreira, F. and Ferreira, G. (2009). Commuting conversions vs. the standard conversions of the “good” connectives. Studia Logica, 92(1), 63–84.Google Scholar
Francez, N. and Dyckhoff, R. (2012). A note on harmony. Journal of Philosophical Logic, 41(3), 613–628.Google Scholar
Hallnäs, L. and Schroeder-Heister, P. (1990). A proof-theoretic approach to logic programming I. Clauses as rules. Journal of Logic and Computation, 1(2), 261–283.Google Scholar
Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik, Berlin: Springer.Google Scholar
Moriconi, E. and Tesconi, L. (2008). On inversion principles. History and Philosophy of Logic, 29(2), 103–113.Google Scholar
Naibo, A. and Petrolo, M. (2015). Are uniqueness and deducibility of identicals the same? Theoria, 81(2), 143–181.Google Scholar
Prawitz, D. (1965). Natural Deduction. A Proof-Theoretical Study, Stockholm: Almqvist & Wiksell. Reprinted in 2006 for Dover Publication.Google Scholar
Prawitz, D. (1971). Ideas and results in proof theory, In Fenstad, J. E., editor, Proceedings of the Second Scandinavian Logic Symposium, Volume 63 of Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, pp. 235–307.Google Scholar
Prawitz, D. (1973). Towards a foundation of a general proof theory, In Suppes, P., Henkin, L., Joja, A., and Moisil, G. C., editors, Proceedings of the Fourth International Congress for Logic, Methodology and Philosophy of Science, Bucharest, 1971, Volume 74 of Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, pp. 225–250.Google Scholar
Prawitz, D. (1979). Proofs and the meaning and completeness of the logical constants, In Hintikka, J., Niiniluoto, I., and Saarinen, E., editors, Essays on Mathematical and Philosophical Logic: Proceedings of the Fourth Scandinavian Logic Symposium and the First Soviet-Finnish Logic Conference, Jyväskylä, Finland, June 29–July 6, 1976, Dordrecht: Kluwer, pp. 25–40.Google Scholar
Prawitz, D. (1994). Review: Michael Dummett, The Logical Basis of Metaphysics. Mind, 103(411), 373–376.Google Scholar
Prior, A. N. (1960). The runabout inference-ticket. Analysis, 21(2), 38–39.Google Scholar
Read, S. (2010). General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic, 39, 557–76.Google Scholar
Schroeder-Heister, P. (1984). A natural extension of natural deduction. The Journal of Symbolic Logic, 49(4), 1284–1300.Google Scholar
Schroeder-Heister, P. (2007). Generalized definitional reflection and the inversion principle. Logica Universalis, 1, 355–376.Google Scholar
Schroeder-Heister, P. (2014). Harmony in proof-theoretic semantics: A reductive analysis, In Wansing, H., editor, Dag Prawitz on Proof and Meaning Volume 7 of Outstanding Contributions to Logic, Heidelberg: Springer, pp. 329–358.Google Scholar
Stålmark, G. (1991). Normalization theorems for full first order classical natural deduction. The Journal of Symbolic Logic, 56(1), 129–149.CrossRefGoogle Scholar
Tennant, N. (1982). Proof and paradox Dialectica, 36, 265–96.Google Scholar
Tranchini, L. (2015a). Proof-theoretic semantics, paradoxes, and the distinction between sense and denotation, Journal of Logic and Computation. DOI: 10.1093/logcom/exu028Google Scholar
Tranchini, L. (2015b). Paradox and inconsistency: Revising Tennant’s distinction through Schroeder-Heister’s assumption rules, In Lolli, G., Panza, M., and Venturi, G., editors, From Logic to Practice, Volume 308 of Boston Studies in the Philosophy and History of Science, Heidelberg: Springer, pp. 111–121.Google Scholar
Widebäck, F. (2001). Identity of proofs, Stockholm: Almqvist & Wiksell.Google Scholar